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Summary Covariate adjustment can improve precision in analysing randomized experiments. With fully observed data, regression adjustment and propensity score weighting are asymptotically equivalent in improving efficiency over unadjusted analysis. When some outcomes are missing, we consider combining these two adjustment methods with the inverse probability of observation weighting for handling missing outcomes, and show that the equivalence between the two methods breaks down. Regression adjustment no longer ensures efficiency gain over unadjusted analysis unless the true outcome model is linear in covariates or the outcomes are missing completely at random. Propensity score weighting, in contrast, still guarantees efficiency over unadjusted analysis, and including more covariates in adjustment never harms asymptotic efficiency. Moreover, we establish the value of using partially observed covariates to secure additional efficiency by the missingness indicator method, which imputes all missing covariates by zero and uses the union of the completed covariates and corresponding missingness indicators as the new, fully observed covariates. Based on these findings, we recommend using regression adjustment in combination with the missingness indicator method if the linear outcome model or missing-completely-at-random assumption is plausible and using propensity score weighting with the missingness indicator method otherwise. 

Abstract Randomized experiments are the gold standard for causal inference and enable unbiased estimation of treatment effects. Regression adjustment provides a convenient way to incorporate covariate information for additional efficiency. This article provides a unified account of its utility for improving estimation efficiency in multiarmed experiments. We start with the commonly used additive and fully interacted models for regression adjustment in estimating average treatment effects (ATE), and clarify the trade-offs between the resulting ordinary least squares (OLS) estimators in terms of finite sample performance and asymptotic efficiency. We then move on to regression adjustment based on restricted least squares (RLS), and establish for the first time its properties for inferring ATE from the design-based perspective. The resulting inference has multiple guarantees. First, it is asymptotically efficient when the restriction is correctly specified. Second, it remains consistent as long as the restriction on the coefficients of the treatment indicators, if any, is correctly specified and separate from that on the coefficients of the treatment-covariate interactions. Third, it can have better finite sample performance than the unrestricted counterpart even when the restriction is moderately misspecified. It is thus our recommendation when the OLS fit of the fully interacted regression risks large finite sample variability in case of many covariates, many treatments, yet a moderate sample size. In addition, the newly established theory of RLS also provides a unified way of studying OLS-based inference from general regression specifications. As an illustration, we demonstrate its value for studying OLS-based regression adjustment in factorial experiments. Importantly, although we analyse inferential procedures that are motivated by OLS, we do not invoke any assumptions required by the underlying linear models. 

Summary Factorial designs are widely used because of their ability to accommodate multiple factors simultaneously. Factor-based regression with main effects and some interactions is the dominant strategy for downstream analysis, delivering point estimators and standard errors simultaneously via one least-squares fit. Justification of these convenient estimators from the design-based perspective requires quantifying their sampling properties under the assignment mechanism while conditioning on the potential outcomes. To this end, we derive the sampling properties of the regression estimators under a wide range of specifications, and establish the appropriateness of the corresponding robust standard errors for Wald-type inference. The results help to clarify the causal interpretation of the coefficients in these factor-based regressions, and motivate the definition of general factorial effects to unify the definitions of factorial effects in various fields. We also quantify the bias-variance trade-off between the saturated and unsaturated regressions from the design-based perspective.